Download Finding the Angle Between Two Lines

6 Topics in Analytic Geometry
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6.1
Lines
Copyright © Cengage Learning. All rights reserved.
Objectives
 Find the inclination of a line.
 Find the angle between two lines.
 Find the distance between a point and a line.
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Inclination of a Line
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Inclination of a Line
In this section, you will look at the slope of a line in terms of
the angle of inclination of the line.
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Inclination of a Line
Every nonhorizontal line must intersect the x-axis. The angle
formed by such an intersection determines the inclination of
the line, as specified in the following definition.
Horizontal Line
Vertical Line
Acute Angle
Obtuse Angle
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Inclination of a Line
The inclination of a line is related to its slope in the
following manner.
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Example 1 – Finding the Inclination of a Line
Find the inclination of (a) x – y = 2 and (b) 2x + 3y = 6.
Solution:
a. The slope of this line is m = 1. So, its inclination is
determined from tan  = 1. Note that m  0. This means
that
 = arctan 1 =  / 4 radian
= 45
as shown in Figure 6.1(a).
Figure 6.1(a)
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Example 1 – Solution
b. The slope of this line is
determined from
means that
as shown in Figure 6.1(b).
cont’d
So, its inclination is
Note that m < 0. This
Figure 6.1(b)
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The Angle Between Two Lines
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The Angle Between Two Lines
When two distinct lines intersect and are nonperpendicular,
their intersection forms two pairs of opposite angles.
One pair is acute and the other pair is obtuse. The smaller
of these angles is the angle between the two lines.
If two lines have inclinations 1 and 2, where 1  2 and
2 – 1   /2, then the angle between the two lines is
 = 2 – 1.
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The Angle Between Two Lines
As shown in Figure 6.2.
Figure 6.2
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The Angle Between Two Lines
You can use the formula for the tangent of the difference of
two angles
tan  = tan(2 – 1)
to obtain the formula for the angle between two lines.
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The Angle Between Two Lines
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Example 2 – Finding the Angle Between Two Lines
Find the angle between 2x – y = 4 and 3x + 4y = 12.
Solution:
The two lines have slopes of m1 = 2 and m2 =
,
respectively. So, the tangent of the angle between the two
lines is
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Example 2 – Solution
cont’d
Finally, you can conclude that the angle is
 = arctan
 1.3909 radians
 79.7
as shown in Figure 6.3.
Figure 6.3
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The Distance Between a
Point and a Line
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The Distance Between a Point and a Line
Finding the distance between a line and a point not on the
line is an application of perpendicular lines.
This distance is defined as the length of the perpendicular
line segment joining the point and the line, as shown below.
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The Distance Between a Point and a Line
Remember that the values of A, B, and C in this distance
formula correspond to the general equation of a line,
Ax + By + C = 0.
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Example 3 – Finding the Distance Between a Point and a Line
Find the distance between the point (4, 1) and the line
y = 2x + 1.
Solution:
The general form of the equation is –2x + y – 1 = 0.
So, the distance between the point and the line is
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Example 3 – Solution
cont’d
The line and the point are shown in Figure 6.4.
Figure 6.4
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